List Multicoloring of Planar Graphs and Related Classes

Abstract

For positive integers $a$ and $b$, a graph $G$ is $(a:b)$-choosable if, for each assignment of lists of $a$ colors to the vertices of $G,$ each vertex can be colored with a set of $b$ colors from its list so that adjacent vertices are colored with disjoint sets. We show that for positive integers $a$ and $b$, every bipartite planar graph is $(a:b)$-choosable iff $\frac{a}{b}\ge 3$. For general planar graphs, we show that if $\frac{a}{b}<4\frac{2}{5}$, then there exists a planar graph that is not $(a:b)$-choosable, thus improving on a result of X. Zhu, which had $4\frac{2}{9}$. Lastly, we show that every $K_5$-minor-free graph is $(a:b)$-choosable iff $\frac{a}{b}\ge 5$. Along the way, we mention some open problems.